Effective field theories forquarkonium at finite temperature

Antonio Vairo

Technische Universitat Munchen

based on

N. Brambilla, J. Ghiglieri, P. Petreczky and A. Vairo

Static quark-antiquark pairs at finite temperature

Phys. Rev. D 78, 014017 (2008)

Outline

1. Motivations

2. Framework

2.1 Scales/effective field theories

2.2 Weak coupling

2.3 Static limit

2.4 Real time

3. Static potential, energy and decay width

3.1 T <∼V

3.2 1/r ≫ T ≫ V

3.3 T ≫ 1/r

4. Conclusions

Motivations

• Thermal medium induces color screening.

• Color screening induces hadron dissociation and quark-gluon plasma formation.Quarkonium dissociation may be a clear and dynamical signature of the formationof this new state of matter.

◦ Matsui Satz PLB 178(86)416

◦ Leitch 08

Free energy vs potential

• Either phenomenological potentials have been used so far or the free energycalculated on the lattice.

• The free energy is not the static potential: the average free energy(∼ 〈TrL†(r)TrL(0)〉) is an overlap of singlet and octet quark-antiquark states,what is called the singlet (∼ 〈TrL†(r)L(0)〉) and the octet (∼ 〈TrL†(r)TrL(0)〉

−1/3 〈TrL†(r)L(0)〉) free energy are gauge dependent;

-200

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

Fi(r,T) [MeV]

r [fm]

Fqq T/Tc=0.91F1 T/Tc=0.91F8 T/Tc=0.91Fqq T/Tc=1.24F1 T/Tc=1.24F8 T/Tc=1.24

◦ Kaczmarek Zantow PRD 71 (2005) 114510

The quarkonium potential at finiteT

In order to study quarkonium properties in a thermal bath at temperature T , the quantityto be determined is the quarkonium potential, which describes the real-time evolution ofa QQ pair through the Schrödinger equation

E Φ =

„

p2

m+ V (r, T )

«

Φ

• In the full theory, V (r, T ) must come from a systematic expansion

• in 1/m (non-relativistic expansion), the leading term being the static potential;

• in the energy E (ultrasoft expansion).

It will encode all contributions from scales larger than E and smaller than m.

• One may exploit these expansions by constructing a suitable hierarchy of EFTs.

• EFTs will account for thermal effects both of the potential and/or of thenon-potential type.

Scales

Quarkonium in a medium is characterized by different energy and momentum scales:

• the scales of the bound state (v is the relative heavy-quark velocity):- m (mass),- mv (momentum transfer, inverse distance),- mv2 (kinetic energy, binding energy, potential V ), ...

• the thermodynamical scales:- T (temperature),- mD (Debye mass, i.e. screening of the chromoelectric interactions), ...

If these scales are hierarchically ordered (if the bound state is non relativistic: v ≪ 1; inthe weak coupling regime T ≫ mD) then we may expand physical observables in theratio of the scales. If we separate/factorize explicitly the contributions from the differentscales at the Lagrangian level this amounts to substituting QCD with a hierarchy ofEFTs, which are equivalent to QCD order by order in the expansion parameters.

Effective Field Theories

QCD

NRQCD

pNRQCDpNRQCD

NRQCDHTL

HTL

m

1/r ∼ mv

V ∼ mv2

TmD

◦ Brambilla Pineda Soto Vairo RMP 77(05)1423

We assume that bound states exist for

• T ≪ m

• 1/r ∼ mv >∼mD

We neglect smaller thermodynamical scales.

Weak coupling

In the weak coupling regime:

• v ∼ αs ≪ 1; valid for tightly bound states: Υ(1S), J/ψ, ...

• T ≫ gT ∼ mD .

Effects due to the scale ΛQCD will not be considered.

Static limit of QCD/NRQCD

We assume m≫ any other scale.

• This allows to integrate out m first and organize the EFTs as expansions in 1/m:the first EFT is NRQCD.

• The leading order term corresponds to the static limit of QCD (or NRQCD):

L = −1

4Fa

µνFa µν +

nfX

i=1

qi iD/ qi + ψ†iD0ψ + χ†iD0χ

ψ (χ) is the field that annihilates (creates) the (anti)fermion.Only longitudinal gluons couple to static quarks.

• The relevant scales in static QCD/NRQCD are: 1/r, V , ... T , mD , ...

Real time

The contour of the partition function is modified to allow for real time:

t0

/Τ

Im t

Re t

it0 -

t f“1”

“2”

In real time, the degrees of freedom double (“1” and “2”), however, the advantages are

• the framework becomes very close to the one for T = 0 EFTs;

• in the static quark sector, the second degrees of freedom, labeled “2”, decouplefrom the physical degrees of freedom, labeled “1”.

This leads to a simpler treatment with respect to alternative calculations in imaginarytime formalism + analytical continuation in real time.

Real-time gluon propagator

• Free gluon propagator in Coulomb gauge:

D(0)00 (~k) =

i

~k2

0

@

1 0

0 −1

1

A

D(0)ij (k) =

„

δij −kikj

~k2

«

8

>

<

>

:

0

B

@

i

k2 + iǫθ(−k0) 2πδ(k2)

θ(k0) 2πδ(k2) −i

k2 − iǫ

1

C

A

+2πδ(k2)nB(|k0|)

0

@

1 1

1 1

1

A

9

=

;

where

nB(k0) =1

ek0/T − 1

In Coulomb gauge, only transverse gluons carry a thermal part.

Real-time static quark propagator

• Free static quark propagator:

S(0)Q (p) =

0

B

@

i

p0 + iǫ0

2πδ(p0)−i

p0 − iǫ

1

C

A

Since [S(0)Q (p)]12 = 0, the static quark fields labeled “2” never enter in any

physical amplitude, i.e. any amplitude that has the physical fields, labeled “1”, asinitial and final states.

Real-time static quark-antiquark propagator

• Free static quark-antiquark propagator:

S(0)

QQ(p) =

0

B

@

i

p0 + iǫ0

2πδ(p0)−i

p0 − iǫ

1

C

A= U

(0)

0

B

@

i

p0 + iǫ0

0−i

p0 − iǫ

1

C

AU

(0)

where

U(0) =

0

@

1 0

1 1

1

A

Similar to the quark propagator, but quark-antiquark fields are bosons.

Real-time potential

• Static quark-antiquark potential:

V =

0

@

V 0

−2i ImV −V ∗

1

A = [U(0)]−1

0

@

V 0

0 −V ∗

1

A [U(0)]−1

Hence the sum of all insertions of a potential exchange between a free quark andantiquark amounts to the full propagator:

U(0)

0

B

@

i

p0 − V + iǫ0

0−i

p0 − V ∗ − iǫ

1

C

AU

(0) = S(0)

QQ(p)

∞X

n=0

h

(−iV(r))S(0)

QQ(p)in

Static quark antiquark atT <∼

V

After having integrated out the scale 1/r the EFT is pNRQCD, which is made of- quark-antiquark states (color singlet S, color octet O),- low energy gluons and light quarks.

The Lagrangian is organized as an expansion in r:

L = −1

4Fa

µνFµν a +

nfX

i=1

qi iD/ qi + Trn

S† (i∂0 − Vs) S + O† (iD0 − Vo)Oo

+VATrn

O†r · gES + S†

r · gEOo

+VB

2Trn

O†r · gEO + O†Or · gE

o

+ · · ·

• At leading order in r, the singlet decouples from the octet and its EOM is:(i∂0 − Vs) S = 0.

• The potentials Vs and Vo are Coulombic: Vs(r) = −CFαs

rand Vo(r) =

αs

2Nc r.

pNRQCD: Feynman rules and loops

θ(T ) e−iTVs θ(T ) e−iTVo

“

e−iR

dt Aadj”

O†r · gES O†{r · gE, O}

• Thermal corrections do not affect the potential, which remains Coulombic, butaffect the static energy and the decay width through loop corrections:

Static quark antiquark atT <∼

V : energy and width

The real part of the diagram gives:

δE =2

3NcCF

α2s

πr T 2 f(Ncαs/(2rT )) , f(z) ≡

Z ∞

0dx

x3

ex − 1P

1

x2 − z2

The imaginary part of the diagram gives

Γ =N3

cCF

6

α4s

rnB(Ncαs/(2r))

• Corrections coming from the scale mD are suppressed by powers of mD/T .

• The width Γ originates from the fact that thermal fluctuations of the medium atshort distances may destroy a color-singlet QQ into an octet plus gluons. Thisprocess is specific of QCD at finite T ; in QCD the relevant diagrams are of the type

Static quark antiquark atT ≪ V

In this limiting case

δE = −8

45π3 CF

Ncr3 T 4 = −

4

3πCF

Ncr3 〈 ~Ea(0) · ~Ea(0)〉T

and

Γ = exponentially suppressed

• δE provides the leading gluon condensate correction to the quark-antiquark staticenergy.

Static quark antiquark at1/r ≫ T ≫ V

Integrating out T from pNRQCD modifies pNRQCD into pNRQCDHTL whose

• Yang–Mills Lagrangian gets the additional hard thermal loop (HTL) part;e.g. the longitudinal gluon propagator at k0 = 0 becomes

i

~k2

0

@

1 0

0 −1

1

A →i

~k2 +m2D

0

@

1 0

0 −1

1

A+ πT

|~k|

m2D

“

~k2 +m2D

”2

0

@

1 1

1 1

1

A

• potentials get an additional thermal correction δV to the Coulomb potential.

Static quark antiquark at1/r ≫ T ≫ V : real part

a)

V

b) ����������������������������

����������������������������

Re δVs(r) =π

9NcCF α2

s r T2 a) ∼ g2r2T 3 ×

V

T

−3

2ζ(3)CF

αs

πr2 T m2

D +2

3ζ(3)NcCF α2

s r2 T 3 b) ∼ g2r2T 3 ×

“mD

T

”2

Static quark antiquark at1/r ≫ T ≫ V : imaginary part

a) Singlet to octet break up contribution

V V

b) ����������������������������

����������������������������

Landau-damping contribution

Im δVs(r) = −N2

cCF

6α3

s T a) ∼ g2r2T 3 ×

„

V

T

«2

+CF

6αs r

2 T m2D

„

1

ǫ+ γE + lnπ − ln

T 2

µ2+

2

3− 4 ln 2 − 2

ζ′(2)

ζ(2)

«

+4π

9ln 2 NcCF α2

s r2 T 3 b) ∼ g2r2T 3 ×

“mD

T

”2

Static quark antiquark at1/r ≫ T ≫ mD ≫ V

Divergences appear in the imaginary part of the potential at order g2r2T 3 דmD

T

”2.

They cancel in physical observables against loop corrections from lower energy scales.

We consider the case 1/r ≫ T ≫ mD ≫ V . Integrating out mD from pNRQCDHTL

leads to an extra contribution δVs to the potential coming from

HTL propagator

Re δVs(r) ∼ g2r2T 3 דmD

T

”3

Im δVs(r) = −CF

6αs r

2 T m2D

1

ǫ− γE + lnπ + ln

µ2

m2D

+5

3

!

Static quark antiquark at1/r ≫ T ≫ mD ≫ V :

energy and width

δE =π

9NcCF α2

s r T2 −

3

2ζ(3)CF

αs

πr2 T m2

D +2

3ζ(3)NcCF α2

s r2 T 3

Γ =N2

cCF

3α3

s T

−CF

3αs r

2 T m2D

2γE − lnT 2

m2D

− 1 − 4 ln 2 − 2ζ′(2)

ζ(2)

!

−8π

9ln 2 NcCF α2

s r2 T 3

• The non-thermal part of the static energy is the Coulomb potential −CFαs/r.

• The thermal width has two origins. The first term comes from the thermal break upof a quark-antiquark color singlet state into a color octet state. The other termscome from imaginary contributions to the gluon self energy that may be tracedback to the Landau damping phenomenon. The first one is specific of QCD, thesecond one would also show up in QED. Having assumed mD ≫ V , the term dueto the singlet to octet break up is parametrically suppressed by (V/mD)2 withrespect to the imaginary gluon self-energy contributions.

Static quark antiquark atT ≫ 1/r ≫ mD

In this situation integrating out T from static QCD leads to static NRQCDHTL, which, atone loop, is static NRQCD with the Yang–Mills Lagrangian supplement by the HTLLagrangian.

Subsequently, integrating out 1/r leads to a specific version of pNRQCDHTL where theCoulomb potential gets corrections from HTL insertions.

Static quark antiquark atT ≫ 1/r ≫ mD: real part

a) Coulomb potential: −CFαs

r

b)������

������ m2

D insertion

Re δVs(r) = −CF

2αs rm

2D b) ∼

αs

r× (rmD)2

Static quark antiquark atT ≫ 1/r ≫ mD: imaginary part

a) Singlet to octet break up contribution

b)������

������ −iπm2

DT/|~k| insertion Landau-damping contribution

Im δVs(r) = −N2

cCF

6α3

s T a) ∼αs

r× (rV )2 × (Tr)

+CF

6αs r

2 T m2D

„

1

ǫ+ γE + lnπ + ln(r µ)2 − 1

«

b) ∼αs

r× (rmD)2 × (Tr)

Static quark antiquark atT ≫ 1/r ≫ mD ≫ V

Divergences appear in the imaginary part of the potential at orderαs

r× (rmD)2 × (Tr).

They cancel in physical observables against loop corrections from lower energy scales.

We consider the case T ≫ 1/r ≫ mD ≫ V . Integrating out mD from pNRQCDHTL

leads to an extra contribution δVs to the potential coming from

HTL propagator

Re δVs(r) ∼ g2r2T 3 דmD

T

”3

Im δVs(r) = −CF

6αs r

2 T m2D

1

ǫ− γE + lnπ + ln

µ2

m2D

+5

3

!

Static quark antiquark atT ≫ 1/r ≫ mD ≫ V :

energy and width

δE = −CF

2αs rm

2D

Γ =N2

cCF

3α3

s T

+CF

3αs r

2 T m2D

„

−2γE − ln(rmD)2 +8

3

«

• The non-thermal part of the static energy is the Coulomb potential −CFαs/r.

• Again the thermal width has two origins. The first term comes from the thermalbreak up of a quark-antiquark color singlet state into a color octet state. The otherterms come from imaginary contributions to the gluon self energy that may betraced back to the Landau damping phenomenon. Having assumed mD ≫ V , theterm due to the singlet to octet break up is parametrically suppressed by (V/mD)2

with respect to the imaginary gluon self-energy contributions.

Quarkonium melting temperature

The quarkonium melts in the medium when

Ebinding ∼ Γ

i.e.

g2

r∼ g2Tm2

Dr2 ln

1

mDr

for 1/r ∼ mg2 and mD ∼ g T

T ∼ mg4/3 (ln 1/g)−1/3

◦ Escobedo Soto arXiv:0804.0691, Laine arXiv:0810.1112

Static quark antiquark atT ≫ 1/r ∼ mD

In this situation integrating out T from static QCD leads to static NRQCDHTL, which, atone loop, is static NRQCD with the Yang–Mills Lagrangian supplement by the HTLLagrangian.

Subsequently, we have to integrate out both 1/r and mD at the same time, by using HTLresummed gluon propagators.

Static quark antiquark atT ≫ 1/r ∼ mD: real part

a)

mass contribution

b) HTL propagators

potential contribution

δE = Re [2δm+ δVs(r)] = −CF αsmD − CFαs

re−mDr a) + b) ∼ αsmD

◦ Gava Jengo PLB 105(81)285

◦ Nadkarni PRD 34(86)3904

Static quark antiquark atT ≫ 1/r ∼ mD: imaginary part

a)

Singlet to octet break up contribution

b)

damping rate of a static quark/antiquark

c) HTL propagators

Landau damping contribution

Γ = −2Im δVs(r) = −N2

cCF

6α3

s T a) ∼ αsmD ×

„

V

mD

«2

×T

mD

+2CF αs T

»

1 −2

rmD

Z ∞

0dx

sin(mDr x)

(x2 + 1)2

–

b) + c) ∼ αsmD ×T

mD≫ αsmD

◦ Pisarski PRD 47(93)5589

◦ Laine Philipsen Romatschke Tassler JHEP 0703(07)054

Static quark antiquark atT ≫ 1/r

• Under the condition 1/r ∼ mD the width is larger by a factor T/mD than thepotential and the bound state dissolves.

• Both Landau damping and singlet to octet break up contribute to the decay width.Parametrically the ratio of the two contributions is proportional to (mD/V )2, henceLandau damping dominates when mD ≫ V and singlet to octet break up whenV ≫ mD . Numerically, the singlet to octet contribution may be large also whenparametrically suppressed:

0.2 0.4 0.6 0.8 1rmD

1

2

3

4

5

G HLandau dampingL�������������������������������������������������������G Hsinglet to octetL

αs = 0.2

αs = 0.3

αs = 0.4αs = 0.5

Γ(Landau damping)/Γ(singlet to octet) vs rmD for different values of αs(1/r)

Conclusions I

• In a framework that makes close contact with modern effective field theories fornon-relativistic bound states at zero temperature, we have studied the real-timeevolution of a static quark-antiquark pair in a medium of gluons and light quarks atfinite temperature.

• For temperatures T ranging from values larger to smaller than the inverse distanceof the quark and antiquark, 1/r, and at short distances, we have derived thepotential between the two static sources, their energy and thermal decay width.

• In the medium, the quarkonium melts at a temperature T ∼ mg4/3 (ln 1/g)−1/3.

Conclusions II

• The derived potential/energy is neither the quark-antiquark free energy nor theinternal energy. It is the real-time potential that describes the real-time evolution ofa quarkonium state in a thermal medium. It encodes all contributions coming frommodes with energy and momentum larger than the binding energy.

• For T < V the potential is the Coulomb potential. For T > V the potential getsthermal contributions.

• Two mechanisms contribute to the thermal decay width: the imaginary part of thegluon self energy induced by the Landau damping phenomenon, and thequark-antiquark color singlet to color octet thermal break up. Parametrically, thefirst mechanism dominates for temperatures such that the Debye mass mD islarger than the binding energy, while the latter dominates for temperatures suchthat mD is smaller than the binding energy.